Int. Fin. Markets, Inst. and Money 21 (2011) 867– 873

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Journal of International FinancialMarkets, Institutions & Money

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Short Communication

Pippenger’s CIP-based solution to the forward-bias puzzle:A rejoinder

Alan King ∗

Department of Economics, University of Otago, PO Box 56, Dunedin 9054, New Zealand

a r t i c l e i n f o

Article history:Received 26 August 2011Accepted 15 September 2011Available online 22 September 2011

JEL classification:E44F31G14G15

Keywords:Forward-bias puzzleCovered interest parityEfficient markets hypothesis

a b s t r a c t

Pippenger (2011a) proposed a solution to the longstandingforward-bias puzzle that attracted several comments, to whichhe has recently replied (Pippenger, 2011b). In this rejoinder it isargued that the points Pippenger raises in defence of his solutiondo not effectively rebut the concerns originally raised. In addi-tion, his model is found to generate puzzling regression resultswhen applied to real-world data. It is shown that these resultsarise because his model’s coefficients represent (potentially biased)estimates of the covered interest parity equation’s coefficients.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The forward-bias puzzle relates to the prediction that, in efficient financial markets with rationalagents, the current forward premium – the difference between the natural log of the forward, ft, andspot, st, exchange rates (both defined in domestic currency per unit of foreign currency terms) – shouldbe an unbiased predictor of the actual future change in the spot rate:1

�st+1 = st+1 − st = ˛0 + ˛1(ft − st) + ut, (1)

where ˛0 = 0, ˛1 = 1 and ut is a white-noise error term.

∗ Corresponding author. Tel.: +64 3 4798686; fax: +64 3 4798174.E-mail address: alan.king@otago.ac.nz

1 Note that (1) becomes the uncovered interest parity (UIP) condition when the forward premium, ft − st , is replaced by theinterest differential, it − i∗t .

1042-4431/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.intfin.2011.09.003

868 A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 867– 873

The puzzle with (1) is that empirical estimates of ˛1 are typically closer to zero than unity and insome cases even negatively signed.

Pippenger (2011a) proposes a solution to this longstanding puzzle that begins by assuming coveredinterest parity (CIP) holds, subject only to errors (et) that may arise because of transaction costs:

st = �0 + �1ft + �2(it − i∗t ) ± et, (2)

where �0 = 0, �1 = 1, �2 = −1 and it − i∗t is the interest rate differential between domestic and foreignrisk-free securities with a maturity date matching that associated with the forward exchange rate.2

After advancing (2) one period, Pippenger subtracts st from both sides and then simultaneously addsand subtracts ft from its right-hand side to produce the following expression:

�st+1 = �0 + �1(ft − st) + �2(ft+1 − ft) + �3(it+1 − i∗t+1) ± et+1, (3)

where �0 = 0, �1 = �2 = 1 and �3 = −1.Pippenger (2011a, p. 299) claims that the omission of the two terms (ft+1 − ft) and (it+1 − i∗t+1)

from (1) is “the econometric source of the forward-bias puzzle”. He goes on to estimate (3), obtainingestimates of �1 that are close to unity; almost exactly so, in fact, when Balke and Wohar’s (1998)dataset is used.

Pippenger’s explanation for the forward-bias puzzle prompted several, largely critical, responses(Baillie, 2011; Chang, 2011; King, 2011; Müller, 2011) to which he has recently replied (Pippenger,2011b). However, his defence of his model does not rebut the substance of the points raised againstit. In particular, when questioning the relevance of the artificially generated data series employed inmy comment, Pippenger (2011b) makes a mistaken assumption about their statistical properties and,when tackling Baillie (2011) and Chang’s (2011) concerns, he focuses on challenging some assumptionsthat are not absolutely essential to their main point. I shall expand on each of these points in Sections2 and 3, respectively. Section 4 demonstrates that Pippenger’s CIP-based approach to solving theforward-bias puzzle has puzzling statistical properties when applied to real (and ‘relevant’) data series.Section 5 explains why this is so and Section 6 concludes.

2. Artificially generated data series

In my original comment I argue that, as (3) is a restatement of the CIP relationship, estimatingit can only provide insight into the empirical validity of CIP. This implied that any other equivalentrestatement of (2), where any two series are used instead of ft and st, would be as empirically valid as(3). Moreover, whenever CIP performs very well – as it does when applied to Balke and Wohar’s (1998)database – it is inevitable that �1 will receive an estimate close to unity, regardless of the series chosenin place of ft and st. This point was demonstrated by replacing ft and st in (3) with three different pairsof artificially generated data series (xit, yit, i = 1, 2, 3). In all three cases, the estimate of �1 obtained isstatistically indistinguishable from that based on actual forward premium data.

In his reply, Pippenger (2011b, p. 635) states that, while there are an infinite number of ways ofrestating CIP, “only a couple of them are relevant for solving the forward-bias puzzle. King’s restate-ment is not. . . .my restatement is.” He specifically objects to restatements based on artificial series onthe grounds that:

“. . . regressing (st+1 − yt) against just (xt − yt) always produces biased results because (ft+1 − xt)is correlated with (xt − yt) by construction while (xt − yt) and (it+1 − i∗

i+1) are uncorrelated byconstruction. As shown in Pippenger (2011a) estimates of [˛1] are not necessarily biased because(ft − st) can be, and is, correlated with (it+1 − i∗

i+1).” (Emphasis original)

It is correct to say that a coefficient estimated for a variable within a model that omits two otherrelevant variables will not necessarily be biased, as the bias attributable to one of the excluded variablesmay offset that attributable to the other under certain conditions, which include the requirement thatboth of the omitted variables be correlated with the retained variable. However, it is not correct to say

2 In the empirical applications below, 30-day interest and forward exchange rates are employed.

A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 867– 873 869

that (xit − yit) and (it+1 − i∗i+1) are uncorrelated by construction. As it happens, in two cases they are,

but (x2t − y2t) and (it+1 − i∗i+1) have a correlation coefficient of almost 0.40.

It is important to note that the point being illustrated by these artificial variables was not that ran-dom series – which should be uncorrelated with (it+1 − i∗

i+1) by construction – could produce resultslargely indistinguishable from those obtained using actual forward and spot rate data, but to illus-trate that any data series, no matter how irrelevant to the issue of forward bias – but not necessarilyuncorrelated with (it+1 − i∗

i+1) – would do so. The choice of artificial over actual data series was madeto ensure they could not be considered proxies for the series they replaced and so bias the results.

3. Pippenger’s restatement of CIP

Pippenger (2011b) claims his restatement is relevant for solving the forward-bias puzzle and hebases its economic interpretation on the variable rt+1, the actual net (or excess) return from an uncov-ered foreign investment. When transaction costs are ignored, rt+1 can be defined as the differencebetween st+1 and ft or, equivalently, the difference between �st+1 and (ft − st) – i.e., the first twovariables in (3).

When CIP holds, rt+1 can also be measured (again, ignoring transaction costs) by the differencebetween (ft+1 − ft) and (it+1 − i∗

i+1) – i.e., the last two variables in (3). However, this implies that (3)– or, to make the point clearer, this slightly rearranged version (with all coefficients assigned theirexpected value):

�st+1 − (ft − st) = (ft+1 − ft) − (it+1 − i∗t+1) ± et+1 (3a)

simply boils down to a regression of one measure of rt+1 against another. The first measure is true bydefinition, whereas the second is true conditional upon CIP being satisfied.

Baillie (2011) and Chang (2011) both make essentially this point, though arguably they overstatetheir case. Specifically, Baillie (2011) considers CIP an identity and Chang (2011) implicitly also doesso when describing (3) as a tautology. As Pippenger (2011b) correctly points out, CIP is a theory thatcould conceivably fail to hold; it is not a relationship that is true by definition, but one that exists whenconditions for covered interest arbitrage are ideal. The fact that such conditions happen to exist in aspecific case does not alter CIP’s status as a theory.

Hence, there is no guarantee that both measures of rt+1 will actually match. How well (3) fits thedata entirely depends on how well the CIP assumption is satisfied. If CIP holds in practice, then (3)contains two equivalent measures of rt+1 and a regression based on it is sure to produce coefficientestimates that reflect their one-to-one relationship.

So, (3) can be used to test a theory, but it is the theory of CIP and not a theory of why UIP fails to hold.This is because neither measure of rt+1 offers an explanation for the existence of the other. They bothmerely record the existence of excess returns on uncovered foreign investments (i.e., that UIP does nothold and so forward rates are biased predictors of future spot rates). Neither measure of rt+1 adds to ourunderstanding as to why investors require such excess returns in the first place or what causes theirsize to change over time. Pippenger (2011b, p. 636), perhaps unwittingly, essentially states as much inconcluding that “. . . future research should concentrate on explaining why the expected return frominvesting without cover is not zero.” This is simply a restatement of the original forward-bias/UIPpuzzle.

4. A puzzle regarding the forward-bias-puzzle solution

As mentioned above, Pippenger (2011b) considers restatements of CIP incorporating artificial datato be irrelevant. This raises the question of whether the CIP-based approach can produce results thatbehave as they should when applied to real-world data. Specifically, will the results (and the estimateof �1 in particular) react appropriately when the data series are modified in ways that have a known(or, at least, predictable) effect on ˛1? The answer, in short, is no.

870 A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 867– 873

Table 1Regression results for Eqs. (4) and (5) (and modified versions thereof).

Eq. (4) Eq. (5)

ˆ̨ 0 ˆ̨ 1 R̄2/DW �̂0 �̂1 �̂2 �̂3 R̄2/DW

[A] Original equationsit (i∗t ) represented by one-month UK(US) interest rates

0.0071(2.979)

−2.0434(−3.087)

0.02710.0777

0.0001(5.342)

0.9928(96.36)

0.9997(3379)

−1.0195(−77.20)

0.99981.5576

[B] Original equations modified by(i) Multiplying all it and i∗t terms by −1 0.0071

(2.979)2.0434(3.087)

0.02710.0777

0.0001(5.342)

1.0066(98.21)

0.9997(3379)

−1.0195(−77.20)

0.99981.5576

(ii) Multiplying all it and i∗t terms by 12(i.e., annualise)

0.0071(2.979)

−0.1703(−3.087)

0.02710.0777

0.0001(5.342)

0.9991(1074)

0.9997(3379)

−1.0195(−77.20)

0.99981.5576

(iii) Multiplying all it , i∗t , it+1 and i∗t+1

terms by 12 (i.e., annualise)0.0071(2.979)

−0.1703(−3.087)

0.02710.0777

0.0001(5.342)

0.9991(1074)

0.9997(3379)

−0.0850(−77.20)

0.99981.5576

(iv) Replacing all it (i∗t ) terms withlong-term UK (US) interest rates

0.0071(3.086)

−3.4202(−3.762)

0.03400.0773

0.0001(5.570)

1.0012(91.29)

0.9997(3357)

−1.0263(−102.8)

0.99981.5612

(v) Replacing all it (i∗t ) terms withGerman (Japanese) interest rates

0.0011(0.526)

0.4693(0.446)

0.00060.0748

0.0001(3.341)

1.0094(195.8)

0.9997(3349)

−1.0244(−166.4)

0.99981.5629

Notes: t-ratios (based on the Newey-West correction method) are shown in parentheses. The sample period is 2 January 1974to 30 September 1993.

To see why, it is first worth reiterating that the forward-bias puzzle and the empirical failure of theUIP hypothesis are the same problem. Any explanation for one must explain the other.3 Therefore, letus begin by re-stating (1) as the UIP hypothesis:

�st+1 = ˛0 + ˛1(it − i∗t ) + vt . (4)

where ˛0 = 0, ˛1 = 1 and vt is a white-noise error term.If we again assume that (2) holds, advance it one period, subtract st from both sides and finally add

and subtract the current interest differential from the right-hand side, we are left with the followingexpression:

�st+1 = �0 + �1(it − i∗t ) + �2(ft+1 − st − it + i∗t ) + �3(it+1 − i∗t+1) ± et+1 (5)

where �0 = 0, �1 = �2 = 1 and �3 = −1.Again we find two variables in (5) not present in (4): namely, the future interest differential, as

before, and (ft+1 − st − it + i∗t ), which has no direct interpretation but under the CIP assumption itreduces to (ft+1 − ft). In addition, (it − i∗t ) is equal to (ft − st) under CIP. Hence, Eqs. (5) and (3) areequivalent expressions; they are identical when CIP is true. Eq. (5) is merely a more convenient vehiclefor considering the effect of modifying the variables introduced when restating the CIP relationship.

Least-squares regression results for Eqs. (4) and (5) based on Balke and Wohar’s (1998) UK–USdataset are presented in Panel A of Table 1.4 In all key respects they are consistent with Pippenger’s(2011a, Tables 1 and 2) original results for Eqs. (1) and (3). Panel B of Table 1 contains the results forfive additional sets of regressions where Eqs. (4) and (5) have been modified by either (i) multiplyingit and i∗t by negative one, (ii) multiplying it and i∗t by 12 (i.e., converting them from rates per monthto rates per annum), (iii) annualising all interest rate variables, (iv) replacing it and i∗t with long-term

3 Unless the explanation pertains to the failure of the CIP assumption inherent in (1), but absent from the UIP hypothesis.This is not the case here.

4 To conserve space, only the results for the full sample period are reported. The results for each sub-period are available onrequest but differ little from those presented in Table 1.

A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 867– 873 871

Table 2Regression results for Eqs. (2a) and (3).

�̂0 �̂1 �̂2 R̄2/DW

Eq. (2a): st+1 = �0 + �1ft+1 + �2(it+1 − i∗i+1

) ± et+1

US–UK: 2 January 1974 to 30 September 1993 0.00004 0.99996 −1.0254 1.0000(1.044) (15360) (−167.0) 1.5612

US–UK: 2 January 1974 to 1 November 1983 0.00007 0.99999 −1.0280 1.0000(0.936) (9038) (−118.1) 1.5361

US–UK: 2 November 1983 to 30 September 1993 0.00001 0.99994 −1.0187 1.0000(0.282) (8820) (−144.3) 1.6058

�̂0 �̂1 �̂2 �̂3 R̄2/DW

Eq. (3): �st+1 = �0 + �1(ft − st) + �2(ft+1 − ft) + �3(it+1 − i∗i+1

) ± et+1

US–UK: 2 January 1974 to 30 September 1993 0.00007 0.99036 0.99970 −1.0171 0.9998(5.258) (102.9) (3358) (−80.63) 1.5557

US–UK: 2 January 1974 to 1 November 1983 0.00008 0.98955 1.0000 −1.0185 0.9997(5.360) (92.85) (1704) (−67.31) 1.5277

US–UK: 2 November 1983 to 30 September 1993 0.00004 0.99180 0.99956 −1.0101 0.9999(2.147) (58.99) (3133) (−54.74) 1.6058

Notes: t-ratios (based on the Newey–West correction method) are shown in parentheses.

interest rates for the UK and the US, respectively, or (v) replacing it and i∗t with short-term interestrates for Germany and Japan, respectively.5

If (ft+1 − st − it + i∗t ) and (it+1 − i∗i+1) truly represent variables whose omission from (4) is respon-

sible for the estimates of ˛1 deviating from its expected value and whose presence in (5) enables �1to conform to that expected value, then the results for (5) shown in Panel B should differ from thosein Panel A in the following ways: (i) �1 = −1; (ii) �1 = 0.0833; (iii) �1 = 0.0833, �3 = −0.0833; (iv) and(v) �0 /= 0, �1, �2 /= 1, �3 /= −1 (i.e., there should be some sign of error-in-variables bias). However,as can be seen, not only do the coefficient estimates not change in the manner expected, they barelychange at all – with the exception of case (iii) where �3 does react as expected when (it+1 − i∗

i+1) isannualised.

The estimates of �1 reported in Table 1 are completely robust to both trivial and fundamentalchanges to the nature of it and i∗t that alter the expected value of ˛1. There is a purely statisticalexplanation for this puzzle: �1 is not, in fact, the unbiased estimate of ˛1 from (4) (or (1) for thatmatter). It is instead a potentially biased estimate of �1 from (2), i.e., the CIP relationship.

5. The forward-bias-puzzle-solution-puzzle’s solution

To see how this is so, let us reconsider the derivation of Pippenger’s solution to the forward-biaspuzzle (3), without (initially, at least) assuming that CIP holds.6 For convenience, we start by restating(2), advanced one period, in general terms:

st+1 = �0 + �1ft+1 + �2(it+1 − i∗t+1) ± et+1. (2a)

5 Daily observations of the one-year sterling interbank lending rate, the yield on ten-year US Treasury secu-rities, the one-month money market rate reported by Frankfurt banks and the official Japanese discount rateare obtained from the Bank of England (http://www.bankofengland.co.uk/statistics/index.htm), the US FederalReserve (http://www.federalreserve.gov/releases/h15/data.htm), the Bundesbank (http://www.bundesbank.de/statistik/statistik zinsen.en.php) and the Bank of Japan (http://www.boj.or.jp/en/statistics/boj/other/discount/index.htm/), respec-tively. Data for the sterling interbank rate only extends back to January 1978 and so the average UK bank base rate is used priorto this date (data accessed 13 August 2011).

6 What follows can as easily be applied to the derivation of Eq. (5); the conclusion will be the same.

872 A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 867– 873

Adding and subtracting ft to and from the right-hand side of (2a) then gives:

st+1 = �0 + ft − (1 − �1)ft + �1(ft+1 − ft) + �2(it+1 − i∗t+1) ± et+1

st+1 = �0 + �1ft + �1(ft+1 − ft) + �2(it+1 − i∗t+1) ± et+1 (6)

Subtracting st from both sides of (6) then gives:

�st+1 = �0 + �1(ft − st) − (1 − �1)st + �1(ft+1 − ft) + �2(it+1 − i∗t+1) ± et+1. (7)

When CIP holds – in particular, when �1 = 1 – the standalone st variable disappears and (7) effectivelyreduces to (3), restated below:

�st+1 = �0 + �1(ft − st) + �2(ft+1 − ft) + �3(it+1 − i∗t+1) ± et+1,

where �0 = �0, �1 = �2 = �1 and �3 = �2.What this reveals about (3) is that both �1 and �2 represent estimates of the same CIP coefficient,

�1. More specifically, �1 and �2 represent potentially biased estimates of �1, as Pippenger’s modeldoes not impose the restriction �1 = �2 and it omits a variable (st) that is relevant whenever �1 is notexactly equal to one.7 Note that, even though they are both estimates of the same coefficient, it ispossible for estimates of �1 and �2 to differ numerically (if not statistically), as their degree of biaswill depend on their associated variable’s correlation with the omitted variable, st (Wooldridge, 2009).However, the stronger CIP holds in a given case, the less scope there would be for this source of bias tohave a noticeable effect. This can be illustrated using Balke and Wohar’s (1998) dataset. As can be seenin Table 2, CIP holds almost perfectly in this instance and there is a very close correspondence betweenthe estimates of �0, �1 and �2 from (2a) and those found for �0, �1 and �2, and �3, respectively, whenestimating (3) – or (5), as can be seen in Panel A of Table 1.

What (7) and the results in Table 2 demonstrate is that the coefficient on the forward premiumvariable in Pippenger’s model is not independent of that on the future change in the forward rate andneither coefficient is independent of that on the forward rate in the CIP relationship. This is why theresults for (5) in Table 1 (and those reported in King, 2011, based on artificial series) are almost entirelyinsensitive to changes in the way it and i∗t are defined. As neither variable appears in (2a), literally anychange in their nature will have no effect on the value of �0, �1 or �2 and hence will have no effect onthe underlying value of �0, �1, �2 or �3, regardless of whether the change alters the expected values of˛0 and ˛1. This is also why the only change seen in the results arises in case (iii), when the definitionof some of the CIP variables (i.e., st+1, ft+1, it+1, i∗t+1) is altered, which affected �2 and hence �3.

No matter how the two variables introduced into the CIP relationship under Pippenger’s approachare defined, the coefficient �1 always represents a (potentially biased) estimate of �1 and will match(subject to differences in the degree of bias) that on ft+1. The tiny changes observed in the parameterestimates for �1 across cases (i)–(v) (vis-á-vis the original estimates for (5)) reported in Table 1 primar-ily reflect differences in the correlation between st and each alternative measure of it and i∗t . In case (i),for example, where it and i∗t are multiplied by −1 (thus reversing the sign of their correlation with st),the estimate of �1 simply switches from being a slight underestimate to being a slight overestimateof �1.

6. Conclusion

Pippenger’s (2011a, 2011b) CIP-based solution to the forward-bias puzzle identifies two variablesthat do not appear in the forward-rate unbiasedness or UIP equations, which he claims can explainwhy these hypotheses fail in practice. However, these variables only represent a way of measuring theexcess returns from uncovered investments that is conditional on CIP being true. They do not explainwhy investors require such excess returns in the first place.

7 For the same reasons, �3 is a potentially biased estimate of the other CIP parameter, �2.

A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 867– 873 873

In addition, when Pippenger’s model generates an estimated coefficient on the forward premium(�1) of unity, this only reflects the existence of a one-to-one relationship between the spot and for-ward rates in the CIP relationship. These estimates of �1 are insensitive to modifications to individualvariables that alter the predicted value of coefficients within the UIP hypothesis. This is because �1 isnot an unbiased estimate of the slope coefficient in the forward-rate unbiasedness or UIP equations,as Pippenger suggests; it is simply a (potentially biased) estimate of the slope coefficient in the CIPrelationship. Consequently, his model reveals nothing about the reasons for the empirical failure ofUIP. Forward bias continues to remain a puzzle.

References

Baillie, R.T., 2011. Possible solutions to the forward bias paradox. Journal of International Financial Markets, Institutions andMoney 21, 617–622.

Balke, N., Wohar, M., 1998. Nonlinear dynamics and covered interest rate parity. Empirical Economics 23, 535–559.Chang, S.S., 2011. On the (in)feasibility of covered interest parity as a solution to the forward bias puzzle. Journal of International

Financial Markets, Institutions and Money 21, 611–616.King, A., 2011. A comment on: the solution to the forward-bias puzzle. Journal of International Financial Markets, Institutions

and Money 21, 623–628.Müller, C., 2011. The forward-bias puzzle: still unsolved. Journal of International Financial Markets, Institutions and Money 21,

605–610.Pippenger, J., 2011a. The solution to the forward-bias puzzle. Journal of International Financial Markets, Institutions and Money

21, 296–304.Pippenger, J., 2011b. The solution to the forward-bias puzzle: reply. Journal of International Financial Markets, Institutions and

Money 21, 629–636.Wooldridge, J.M., 2009. Introductory Econometrics. South-Western, Mason, OH.